Lecture 1
However, the macro economy is made up of millions of micro decisions.
Markets coordinate these decisions invisibly and often effectively.
However, sometimes markets produce undesirable outcomes or fail.
In this course, we will study:
Central question: How do individuals make choices?
Economic Approach to Choice
Does everyone really “maximize utility”?
We consider a consumer choosing between bundles of goods.
Key question: What properties should preferences satisfy for them to be “rational”?
For any two bundles \(A\) and \(B\), the consumer can state which is preferred or that they are indifferent: \[A \succeq B, \quad B \succeq A, \quad \text{or both (indifference)}\]
Interpretation: Consumers can always make comparisons. Rules out indecision.
If \(A \succeq B\) and \(B \succeq C\), then \(A \succeq C\)
Interpretation: Preferences are internally consistent. No cycles.
Small changes in consumption bundles lead to small changes in preferences.
Technical: For any bundle \(A\), the sets \(\{B : B \succeq A\}\) and \(\{B : A \succeq B\}\) are closed.
Interpretation: No sudden jumps. Preferences are “smooth.”
More is better: If \(A\) has at least as much of everything as \(B\), and strictly more of at least one good, then \(A \succ B\).
Interpretation: Consumers always prefer more to less (at least weakly).
Averages are preferred to extremes. If \(A \sim B\), then: \[\lambda A + (1-\lambda)B \succeq A \text{ for } \lambda \in [0,1]\]
Interpretation: Consumers prefer balanced consumption bundles. Diminishing marginal rate of substitution.
Example: If you’re indifferent between (5 apples, 0 oranges) and (0 apples, 5 oranges), you prefer (5 apples, 5 oranges) to either extreme.
Framing effects: Preferences change based on how options are presented
Intransitivity: Preference reversals in complex choices (Condorcet paradox)
Present bias: Time-inconsistent preferences (\(\beta\delta\) model)
Reference dependence: Preferences depend on current endowment (loss aversion)
Bounded rationality
Key Theorem: If preferences satisfy completeness, transitivity, continuity, and monotonicity, then there exists a continuous utility function \(U(x,y)\) that represents them: \[A \succeq B \iff U(A) \geq U(B)\]
Interpretation: We can assign numbers to bundles such that higher numbers = more preferred.
Important: Utility is ordinal, not cardinal
Since utility is ordinal, we can apply any strictly increasing transformation without changing preferences:
If \(U(x,y)\) represents preferences, so does \(V(x,y) = f(U(x,y))\) for any strictly increasing \(f\).
Examples:
Why this matters: We can transform utility functions to make calculations easier.
An indifference curve is the set of all bundles that give the same utility level:
\[IC(U_0) = \{(x,y) : U(x,y) = U_0\}\]
Interpretation: The consumer is indifferent between any two points on the same curve.
Under our axioms, indifference curves must be:
Downward sloping (from non-satiation)
Do not cross (from transitivity)
Convex to the origin (from convexity of preferences)
Higher curves represent higher utility (from monotonicity)
The marginal rate of substitution is the rate at which the consumer is willing to trade good \(Y\) for good \(X\) while maintaining constant utility.
Geometrically: MRS = -(slope of indifference curve)
\[MRS = -\frac{dy}{dx}\bigg|_{U=const}\]
Interpretation: How many units of \(Y\) are you willing to give up to get one more unit of \(X\)?
Example: If MRS = 2, you’re willing to give up 2 units of \(Y\) to get 1 more unit of \(X\) (and remain indifferent).
Along an indifference curve, utility is constant: \(U(x,y) = \bar{U}\)
Taking the total differential: \[dU = \frac{\partial U}{\partial x}dx + \frac{\partial U}{\partial y}dy = 0\]
Rearranging: \[\frac{\partial U}{\partial y}dy = -\frac{\partial U}{\partial x}dx \quad \rightarrow \quad \frac{dy}{dx} = -\frac{\partial U/\partial x}{\partial U/\partial y} = -\frac{MU_x}{MU_y}\]
Therefore: \[\boxed{MRS = -\frac{dy}{dx} = \frac{MU_x}{MU_y}}\]
Convexity assumption → Diminishing MRS
As you consume more of good \(X\) (moving right along an IC), the MRS decreases:
Economic intuition: Scarcity increases value. The less you have of something, the more you value additional units.
Exercise: Consider the utility function \(U(x,y) = x^{0.3}y^{0.7}\)
Hint: \(MU_x = \frac{\partial U}{\partial x}\), and remember \(MRS = \frac{MU_x}{MU_y}\)
Goods that can be substituted at a constant rate.
\[U(x,y) = ax + by\]
Examples:
Key features:
Goods that must be consumed in fixed proportions.
\[U(x,y) = \min\{ax, by\}\]
Examples:
Key features:
The most widely used functional form in economics:
\[U(x,y) = x^{\alpha}y^{\beta}\]
Or equivalently (applying monotonic transformation):
\[U(x,y) = \alpha \ln x + \beta \ln y\]
Key features:
For \(U(x,y) = x^{\alpha}y^{\beta}\):
Step 1: Find marginal utilities \[MU_x = \frac{\partial U}{\partial x} = \alpha x^{\alpha-1}y^{\beta}\] \[MU_y = \frac{\partial U}{\partial y} = \beta x^{\alpha}y^{\beta-1}\]
Step 2: Calculate MRS \[MRS = \frac{MU_x}{MU_y} = \frac{\alpha x^{\alpha-1}y^{\beta}}{\beta x^{\alpha}y^{\beta-1}} = \frac{\alpha}{\beta} \cdot \frac{y}{x}\]
MRS depends on the ratio \(y/x\) and the preference parameters \(\alpha/\beta\).
Constant Elasticity of Substitution (CES) utility function:
\[U(x,y) = (ax^{\rho} + by^{\rho})^{1/\rho}, \quad \rho \leq 1, \rho \neq 0\]
Elasticity of substitution: \(\sigma = \frac{1}{1-\rho}\)
Special cases:
Flexibility: CES nests all three cases mentioned.
Consumers have limited income \(I\) and face prices \(p_x, p_y\) for goods:
\[p_x \cdot x + p_y \cdot y \leq I\]
Budget line: Set of bundles that cost exactly \(I\)
\[p_x \cdot x + p_y \cdot y = I\]
Rearranging for \(y\):
\[y = \frac{I}{p_y} - \frac{p_x}{p_y}x\]
The consumer chooses \((x,y)\) to:
\[\max_{x,y} \quad U(x,y)\]
subject to:
\[p_x x + p_y y = I\] \[x \geq 0, \quad y \geq 0\]
Goal: Find the highest indifference curve that touches the budget line.
Intuition: Get as much utility as possible given your budget.
At the tangency point: MRS = \(p_x/p_y\)
Intuition: Consumer’s subjective tradeoff (MRS) equals market tradeoff
If MRS \(>\) \(p_x/p_y\):
If MRS \(<\) \(p_x/p_y\):
The consumer’s problem: \[\max_{x,y} \quad U(x,y) \quad \text{subject to} \quad p_x x + p_y y = I\]
Lagrangian: \[\mathcal{L}(x,y,\lambda) = U(x,y) + \lambda(I - p_x x - p_y y)\]
where \(\lambda\) is the Lagrange multiplier.
First-order conditions (FOCs):
From the first two FOCs: \[MU_x = \lambda p_x \quad \text{and} \quad MU_y = \lambda p_y\]
Dividing these: \[\frac{MU_x}{MU_y} = \frac{p_x}{p_y}\]
This is exactly the tangency condition: MRS = price ratio!
From FOCs: \(\lambda = \frac{MU_x}{p_x} = \frac{MU_y}{p_y}\)
λ = marginal utility of income
Example: If \(\lambda = 0.5\):
Note: λ decreases as income increases (diminishing marginal utility of income)
Setup: \(U(x,y) = x^{\alpha}y^{\beta}\), budget: \(p_x x + p_y y = I\)
Step 1: Form the Lagrangian \[\mathcal{L} = x^{\alpha}y^{\beta} + \lambda(I - p_x x - p_y y)\]
Step 2: Take FOCs \[\frac{\partial \mathcal{L}}{\partial x} = \alpha x^{\alpha-1}y^{\beta} - \lambda p_x = 0\] \[\frac{\partial \mathcal{L}}{\partial y} = \beta x^{\alpha}y^{\beta-1} - \lambda p_y = 0\] \[\frac{\partial \mathcal{L}}{\partial \lambda} = I - p_x x - p_y y = 0\]
Step 3: Combine first two FOCs \[\frac{\alpha x^{\alpha-1}y^{\beta}}{\beta x^{\alpha}y^{\beta-1}} = \frac{p_x}{p_y}\]
Simplifying: \[\frac{\alpha}{\beta} \cdot \frac{y}{x} = \frac{p_x}{p_y}\]
Solving for \(y\): \[y = \frac{\beta}{\alpha} \cdot \frac{p_x}{p_y} \cdot x\]
Step 4: Substitute into budget constraint \[p_x x + p_y \cdot \frac{\beta}{\alpha} \cdot \frac{p_x}{p_y} \cdot x = I\] \[p_x x + \frac{\beta}{\alpha} p_x x = I\] \[p_x x \left(1 + \frac{\beta}{\alpha}\right) = I\] \[p_x x \cdot \frac{\alpha + \beta}{\alpha} = I\]
Step 5: Solve for \(x^*\): \[x^* = \frac{\alpha I}{(\alpha + \beta)p_x}\]
Substitute back to get \(y^*\): \[y^* = \frac{\beta}{\alpha} \cdot \frac{p_x}{p_y} \cdot x^* = \frac{\beta}{\alpha} \cdot \frac{p_x}{p_y} \cdot \frac{\alpha I}{(\alpha + \beta)p_x} = \frac{\beta I}{(\alpha + \beta)p_y}\]
\[\boxed{x^* = \frac{\alpha I}{(\alpha + \beta)p_x}, \quad y^* = \frac{\beta I}{(\alpha + \beta)p_y}}\]
Key results:
Cobb-Douglas utility implies constant expenditure shares regardless of income or prices.
Which of the following goods do you think have roughly constant expenditure shares in real life?
Engel’s Law: As income rises, the proportion spent on food decreases.
How to model this?
Homogeneity of degree zero: If all prices and income are multiplied by the same factor \(t > 0\): \[x^*(tp_x, tp_y, tI) = x^*(p_x, p_y, I)\]
Interpretation: Demand depends only on relative prices, not absolute price level.
Example: If all prices and income double (inflation), consumption bundles don’t change.
Why? The budget constraint becomes: \[tp_x \cdot x + tp_y \cdot y = tI\] \[\Rightarrow p_x \cdot x + p_y \cdot y = I\]
The indirect utility function \(V(p_x, p_y, I)\) gives maximum achievable utility as a function of prices and income:
\[V(p_x, p_y, I) = U(x^*(p_x, p_y, I), y^*(p_x, p_y, I))\]
Interpretation: Maximum utility you can achieve given market conditions.
Properties:
For \(U(x,y) = x^{\alpha}y^{\beta}\):
Substituting \(x^* = \frac{\alpha I}{(\alpha+\beta)p_x}\) and \(y^* = \frac{\beta I}{(\alpha+\beta)p_y}\):
\[V(p_x, p_y, I) = \left(\frac{\alpha I}{(\alpha+\beta)p_x}\right)^{\alpha} \left(\frac{\beta I}{(\alpha+\beta)p_y}\right)^{\beta}\]
Simplifying: \[V(p_x, p_y, I) = \frac{I^{\alpha+\beta}}{p_x^{\alpha}p_y^{\beta}} \cdot \frac{\alpha^{\alpha}\beta^{\beta}}{(\alpha+\beta)^{\alpha+\beta}}\]
Note: \(V\) increases with \(I\) and decreases with \(p_x\) and \(p_y\) ✓
Indirect utility is the tool for welfare analysis of price/income changes
Example: Government raises gas tax by $0.50/gallon. To compensate, gives everyone $200 cash transfer.
Policy question: Give $200 cash or $200 food stamps?
Setup:
With cash: Budget is \((p_x, p_y, I + 200)\)
With food stamps: Can buy up to \(200/p_x\) extra food, but must spend at least that on food
Lump sum principle implies:
Then why do governments use in-kind transfers like SNAP, housing vouchers, etc.?
Similar reasoning can be used to conclude that price subsidies are inferior to cash transfers.
What we covered: